Whenever they ask to get then.

Them*

When they know (and the people around them agree) that they are responsible, and understand exactly what the outcomes are. You might want a piercing. Be dead set on it, then once done, you don't like the look or the feel, or people's responses to it. Or in the professional sense, you might want to work for a serious company. You dye your hair multi or bright colours, employers may not want to invest in someone like that thinking they are not up for the job for different reasons. Don't risk it.
My opinion, definitely in the teenage years upwards to whatever age you feel like doing it. You live one life, and if you know it will make you happy, then go ahead. But think about the people around you, the consequences, and whether or not you really will be happy with it. Might just be a phase at one point so make sure.

Hair dyed, I guess whenever and as far as piercings teens maybe late teens. If you plan on gauging think about if for a while because that's harder to get rid of than say a regular piercing

Agree with what dzrosex said!

Like others have said, its not so much a numerical age as it is mental maturity. So many people make decisions because they're "right" and then later they regret it. Lots of kids are actually thankful people hold them back from most things. My rule is to mull it over for a few months to a year (depending on what it is) and then if ya still want it, get it. Once you're an adult though you have to be more careful. I've put off dying my hair a blue tinted color because I need a new job. Same with tattoos and piercings.